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If you've ever considered how likely something is to occur, you were thinking about odds and probability without even realizing it. Probability is a field of mathematics dealing with how likely something is to happen. The most straightforward example is flipping a fair coin. The only possible outcomes are heads and tails, which have a one-in-two (or $\frac{1}{2}$ or 50%) chance of coming up on any given flip.

Rolling a six-sided die provides a slightly more complicated example. The numbers $\left(1,2,3,4,5,6\right\}$ are all possible outcomes, each having a one-in-six chance of coming up. Probabilities like this are often stated in terms of odds, or the ratio of favorable outcomes to unfavorable outcomes.

Theoretical probability refers to the odds of something happening given infinite instances. For example, if you rolled the six-sided die above an infinite number of times you would expect to get each number $\frac{1}{6}$ of the time. Of course, it's wildly impractical to roll the die an infinite number of times.

Therefore, you might be asked about experimental probability or the observed number of outcomes based on a set number of trials. If you rolled the die five times and got a 3 on four of them, the experimental probability of rolling a 3 was $\frac{4}{5}$ or four out of five even though the theoretical probability is just 1-in-6. Generally speaking, the experimental probability gets closer to the theoretical probability the more often an event is repeated. However, the underlying odds never change. The odds of rolling a 3 the next time remain 1-in-6 even if you've rolled 10 in a row.

Working with probability can be challenging because some problems require you to add probabilities together while others require multiplication instead. For instance, you would add if you were asked about the odds of rolling an even number on a six-sided die because a 2, a 4, or a 6 would all satisfy the criteria. You would take each event's $\frac{1}{6}$ chance of occurring and add them for $\frac{3}{6}$ , a fraction that should then be simplified to $\frac{1}{2}$ . Adding fractions is very important in probability.

However, if you wanted to calculate the odds of rolling a 2 and then a 4, you would multiply each event's independent $\frac{1}{2}$ odds for the combined odds of $\frac{1}{36}$ . Understanding how to multiply fractions is also essential in probability.

If you're ever unsure about whether to add or multiply, consider the keywords present in the problem. If it includes "either", "or", "at least", or a synonym, that means the combined odds will be greater than the independent odds and you should add. If you see keywords like "and", "both", "all", or a synonym, that means the combined odds will be less than the independent odds and you should multiply.

a. What are the odds of rolling a number lower than 5 on a standard six-sided die?

There are 4 numbers lower than 5 on a six-sided die (1, 2, 3, and 4). There are 6 possible outcomes. The odds are:

$\frac{4\left(\text{favorable outcomes}\right)}{6\left(\text{total outcomes}\right)}=\frac{4}{6}=\frac{2}{3}$

So, the odds are two-in-three or $\frac{2}{3}$ .

b. When flipping a fair coin, what are the odds of flipping heads three times in a row?

Each flip has 2 possible outcomes (heads or tails). The probability of flipping heads once is $\frac{1}{2}$ . To find the probability of flipping heads three times in a row, multiply the probabilities:

$\frac{1}{2}\times \frac{1}{2}\times \frac{1}{2}=\frac{1}{8}$

So, the odds are one-in-eight or $\frac{1}{6}$ .

c. What is the theoretical probability of rolling an odd number on a standard six-sided die?

There are 3 odd numbers (1, 3, and 5) on a six-sided die, and there are 6 possible outcomes. The probability is:

$\frac{3\left(\text{favorable outcomes}\right)}{6\left(\text{total outcomes}\right)}=\frac{3}{6}=\frac{1}{2}$

So, the odds are one-in-two or $\frac{1}{2}$

d. Roll a six-sided die 10 times and record the results. What was the experimental probability of rolling an odd number?

Count the number of times you got an odd number (let's say N_odd), then pide it by 10 (the number of total die rolls):

$\text{Experimental probability}=\frac{{N}_{\text{odd}}}{10}$

For example, if you rolled an odd number 6 times, the experimental probability would be $\frac{6}{10}=\frac{3}{5}$ .

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Common Core: 7th Grade Math Diagnostic Tests

Many students don't realize how often odds and probability come up in daily life, providing plenty of practical ways to incorporate them into their lives. For example, video game fans might be interested in the odds of a monster dropping an item while sports fans might look at the probability of kicking a field goal or scoring a goal from a given spot. Private instruction allows learners of all ability levels to receive personalized attention, so contact the Educational Directors at Varsity Tutors today to learn more about how tutoring could deepen your student's understanding of the odds.

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